Hashing - resolving collisions

1
Hashing - resolving collisions

• Open addressing linear probing
• hash function (h(k)i)mod m for i0, 1,...,m-1
• Insert start with the location where the key
  hashed and do a sequential search for an empty
  slot.
• Search start with the location where the key
  hashed and do a sequential search until you
  either find the key (success) or find an empty
  slot (failure).
• Delete (lazy deletion) follow same route but
  mark slot as DELETED rather than EMPTY, otherwise
  subsequent searches will fail (why?).

2
Hashing - resolving collisions

• Open addressing linear probing
• Disadvantage primary clustering
• long sequences of used slots build up with gaps
  between them.
• result performance near sequential search

3
Hashing - resolving collisions

• Open addressing quadratic probing
• Probe the table at slots (h(k)i2) mod m
  for i0, 1,2, 3, ..., m-1
• Careful for some values of m (hash table size),
  very few slots end up being probed.
• Try m16
• It is possible to probe all slots for certain ms.
• For prime m, we get pretty good results.
• if the table is at least half-empty, an element
  can always be inserted

4
Hashing - resolving collisions

• Open addressing quadratic probing
• Better than linear probing but may result in
  secondary clustering if h(k1)h(k2) the probing
  sequences for k1 and k2 are exactly the same
• general hash function (h(k)c1ic2i2) mod m

5
Hashing - resolving collisions

• Open addressing double hashing
• The hash function is (h(k)i h2(k)) mod m
• In English use a second hash function to obtain
  the next slot.
• The probing sequence is
• h(k), h(k)h2(k), h(k)2h2(k), h(k)3h3(k),
  ...
• Performance
• Much better than linear or quadratic probing.
• Does not suffer from clustering
• BUT requires computation of a second function

6
Hashing - resolving collisions

• Open addressing double hashing
• The choice of h2(k) is important
• It must never evaluate to zero
• consider h2(k)k mod 9 for k81
• The choice of m is important
• If it is not prime, we may run out of alternate
  locations very fast.
• If m and h2(k) are relatively prime, well end
  up probing the entire table.
• A good choice for h2 is h2(k)p- (k mod p) where
  p is a prime less than m.

7
Hashing - resolving collisions

• Open addressing random probing
• Use a pseudorandom number generator to obtain the
  sequence of slots that are probed.

8
Hashing - resolving collisions

• Open addressing expected of probes
• Assuming uniform hashing...
• Insert/Unsuccessful search 1/(1-?)
• Successful search (1ln(1/(1- ?))/ ?

9
Hashing - resolving collisions

• Example
• m 13
• sequence of keys 18-26-35-9-64-47-96-36-70
• h1(k) k mod 13
• Insert the sequence into a hash table using
• linear probing
• quadratic probing
• double hashing with h2(k) k mod 7 6

10
Hashing - rehashing

• If the table becomes too full, its performance
  falls.
• The O(1) property is lost
• Solution
• build a bigger table (e.g. approximately twice as
  big) and rehash the keys of the old table.
• When should we rehash?
• when the table is half full?
• when an insertion fails?
• when a certain load factor has been reached?